I stayed at two different hotels in Rome, one not far from Termini, and one about a block from the entrance to the Vatican Museum, which I highly recommend. It's the Hotel Alimandi. Great ambiance, great breakfast buffet, within walking distance of the Vatican. A grocery store and laundromat a block away. A public market about two blocks away.

You probably wouldn't consider the Vatican area to be centrally located, but Rome is a great walking town and has abundant transportation options.

thanks again to all for the replies. I think I'm going to try to book Albergi Cesari, since I keep hearing it suggested over and over again. I was afraid it might not be very clean after reading some of the tripadvisor reviews, but when I emailed them the rates are fabulous so I guess I'll chance a dirty bathroom for the good location and excellent price! ;-)